Die folgenden Vorträge haben im Wintersemester 2012/13 im Oberseminar reelle Geometrie und Algebra, im Oberseminar Modelltheorie und im Schwerpunktskolloquium reelle Geometrie und Algebra stattgefunden.

 

Montag, 22.10.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Kein Vortrag

Freitag, 26.10.2012 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Rainer Sinn (Konstanz)

The algebraic boundary of convex semi-algebraic sets and dualities

Abstract: We will study the algebraic boundary of a semi-algebraic set, i.e. the Zariski closure of its boundary in the usual, euclidean topology, in terms of algebraic and convex duality. The relationship between these two duality theories for convex semi-algebraic sets seems to be close and complicated. We will see examples and even pictures.

Montag, 29.10.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Kein Vortrag

Mittwoch, 31.10.2012 um 14:00 Uhr (im Raum L914), Oberseminar Reelle Geometrie und Algebra

Cordian Riener (Helsinki)

Symmetric nonnegative forms and sum of squares

Abstract: I want to present results on the relationship between nonnegative forms and symmetric sums of squares and the asymptotic behaviour when the degree 2d is fixed and the number of variables n grows. As a first result I will show that in sharp contrast to the general case the difference between symmetric forms and sums if squares does not grow arbitrarily large for any fixed degree 2d. In particular for degree 4 it can be shown that the difference between symmetric nonnegative forms and sums of squares asymptotically goes to 0. More precisely I will relate nonnegative symmetric forms to symmetric mean inequalities, valid independent of the number of variables. Then given a symmetric quartic the related symmetric mean inequality holds for all n ≥ 4, if and only if the symmetric mean inequality can be written as a sum of squares. If time permits I will also try to outline very recent ideas on how this result can be proven for a general degree 2d.

 

Montag, 05.11.2012 um 15.15 Uhr (im Raum D436), Oberseminar Modelltheorie

Samuel Volkweis Leite (Konstanz)

Divisibilities and an Axiomatization of the Ring of Continuous Functions from a Compact Hausdorff Space to the Field of p-adic Numbers

Abstract: I will talk about divisibilities as being the pre-concept of  valuations when treated axiomatically. As an application, I will use them to give an axiomatization of a certain class of rings of continuous functions having p-adic values.

In the last talk I used some stronger condition to give such characterization. Now I will show how we can avoid it using a more reasonable condition.

 

Donnerstag, 08.11.2012 um 17:00 Uhr, Oberseminar Modelltheorie

Tamara Servi (CMAF, Universidade de Lisboa)

(Gast von Salma Kuhlmann und Margaret Thomas)

Quantifier Elimination and Rectilinearisation Theorem for quasi-analytic algebras

Abstract: An algebra of real functions is quasi-analytic if there is an injective morphism which associates a (divergent) generalised power series to each germ at zero of a function in the algebra. In the context of quasi-analytic algebras, which generalises that of real analytic geometry, we prove an analogue to Denef and van den Dries' Quantifier Elimination Theorem and analogue to Hironaka's Rectilinearisation Theorem, which states that every bounded subanalytic set can be written, after a finite sequence of blow-ups, as a finite union of quadrants (joint work with J.-P. Rolin).

Freitag, 09.11.2012 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Mehdi Ghasemi (Konstanz)

An application of Jacobi's theorem to locally multiplicatively convex topological real algebras

Abstract: Let A be a commutative unital algebra over real numbers and let τ be a locally multiplicatively convex topology on A. We apply T. Jacobi’s representation theorem to determine the closure of a ΣA^2d-module S of A in the τ-topology, for any integer d ≥ 1. We show that this closure is exactly the set of all elements a ∈ A such that α(a) ≥ 0 for every τ-continuous R-algebra homomorphism α : A → R with α(S) ⊆ [0,∞). We obtain a representation of any linear functional L : A → R which is continuous with respect to any such τ and nonnegative on S as integration with respect to a unique Radon measure on the space of all real valued R-algebra homomorphisms on A.

This is a joint work with S. Kuhlmann and M. Marshall, which is a continuation to the followings:

With S. Kuhlmann, Closure of the cone of sums of 2d-powers in commutative real topological algebras, to appear in J. Funct. Anal.

With S. Kuhlmann and E. Samei, The moment problem for continuous positive semidefinite linear functionals, to appear in Arch. Math. (Basel).

Montag, 12.11.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Merlin Carl (Konstanz)

On Machines and Melodies - An Introduction to Infinitary Computations

Abstract: Turing computability provides an intuitive and attractive formal framework for the analysis of the question which mathematical objects and functions can be generated by applying a certain procedure finitely many times. However, in mathematical practice, we often encounter examples of objects generated by infinite processes involving limits, such as adding reals, computing with limits of real sequences, forming direct limits of directed systems, hulls, closures etc. This motivates the study of infinitary computations that can account for the limit phenomenon in such processes and should provide a framework for extending results from recursion theory to infinitary mathematical objects. I will introduce Infinite Time Register Machines (ITRM’s), a machine model for infinitary computations introduced by Koepke and Miller in 2008 and give some of the most interesting results that we have about their behaviour and strength. In particular, I will demonstrate the strong link between ITRM- computability and models of Kripke-Platek set theory established by Koepke and Miller and give some results from my recent work on the question which reals can be uniquely identified by an ITRM as the number in the oracle.

Freitag, 16.11.2012 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Christoph Hanselka (Konstanz)

Characteristic Polynomials of Symmetric Matrices

Abstract: The general question in this talk will be, which univariate polynomials with coefficients in some commutative ring R can be expressed as the characteristic polynomial of some symmetric matrix over that ring. Besides some results for R being a field or the ring of integers, the main focus will lie on the univariate polynomial ring over the real numbers. This approach will yield a more algebraic proof of the Helton-Vinnikov Theorem about the existence of determinantal representations of hyperbolic polynomials. Ingredients are Witt's Local Global Principle and a Cassels-Pfister theorem for Central Simple Algebras.

 

Montag, 19.11.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Kein Vortrag

Freitag, 23.11.2012 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Mehdi Ghasemi (Konstanz)

Computing lower bound for minimum of a polynomial using geometric programming

Abstract: We use a result of Hurwitz and Reznick and a result of Fidalgo and Kovacec to give a new sufficient condition for a form to be a sum of squares. We apply this result to obtain a new lower bound f_gp for an even degree polynomial f , and explain how f_gp can be computed using geometric programming. We extend this method to obtain a lower bound for any polynomial f on certain compact semialgebraic sets.

This talk presents parts of a joint work with J. B. Lasserre and M. Marshall and the following articles:

With M. Marshall, Lower bounds for polynomials using geometric programming, SIAM J. Optim. 22(2) (460-473), 2012.

With M. Marshall, Lower bounds for a polynomial in terms of its coefficients, Arch. Math. (Basel) 95 (343-353), 2010.

Montag, 26.11.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Isabella Fascitiello (Seconda Università degli Studi di Napoli)

(Gast von Salma Kuhlmann)

On the construction of a Normal Integer Part of a Real Closed Field

Abstract: An integer part (IP) for an ordered field R is a discretely ordered subring Z such that for each r in R there exists some z in Z with  |r - z|< 1. Shepherdson in 1964 showed that Integer Parts of real closed fields (RCF) are precisely the models of a fragment of Peano Arithmetic called Open Induction (OI). This correspondence led Shepherdson and others to investigate the properties of Integer Parts of real closed fields. By Mourgues and Ressayre, every real closed field has an IP. The IPs they constructed are not normal,  i. e. not integrally closed in its fraction field. It is an open question whether every RCF has a normal IP.

Freitag, 30.11.2012 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Jan Limbeck (Universität Passau)

(Gast von Markus Schweighofer)

Linking exact and approximate border bases

Abstract: Border bases, a generalization of Gröbner bases, have attracted the interest of many researchers recently. Their theoretical aspects have been studied extensively by Mourrain, Robbiano, and Kreuzer. One of the reasons is the enhanced numerical stability of border bases, which makes them suitable also for industrial applications. This was demonstrated by Kreuzer et. al in "Approximate computation of zero-dimensional polynomial ideals" who extended exact border bases to approximate border bases, and who also gave an efficient algorithm (AVI) for their computation. Approximate border bases have been investigated further and successfully applied in the Algebraic Oil Research Project, a collaboration between the University of Passau and Shell Global Solutions. Despite the numerous benefits of approximate border bases, one open problem is the unavailability of important tools of algebra and algebraic geometry for them. A possible solution to bridge this gap is to construct "close by" exact border bases, which turns out to be quite challenging if runtime is a major concern.

In this talk we will initially discuss the basic properties of border bases and outline the key differences to Gröbner bases. Afterwards the concept of approximate border bases will be introduced and its usefulness will be demonstrated on a real world example. Finally, we discuss the challenges and open problems which are associated with approximate border bases, in particular the so-called rational recovery problem, which is concerned with computing close by exact border bases for a given approximate one, and how we hope to address it.

 

Montag, 03.12.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Will Anscombe (University of Oxford)

(Gast von Arno Fehm)

Definability in power series fields via a lemma like Hensel's

Abstract:We study definability in the power series field F((t)) in the language of rings allowing parameters from F. We assume F is perfect. After proving a `Hensel-like Lemma', we show how to give a very simple description of orbits of singletons under F-automorphisms of F((t)); in fact these orbits are existentially-definable (with additional parameters). Thus we may use previous work about existential definability to obtain consequences for F-definable subsets of F((t)).

Freitag, 07.12.2012 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Jochen Koenigsmann (University of Oxford)

(Gast von Salma Kuhlmann)

Definable henselian valuations

Abstract: There has been a growing interest recently in finding first-order definitions in the language of rings for valuations on a field, in particular for henselian valuations. The motivations for this range from model theory over number theory to algebraic geometry. The simpler the definition the better - the best being purely existential or purely universal and parameter-free; and for families of valuations uniformity of the definitions features prominently. There has been a bouquet of nice results (by all means not all my own) which I would like to present. Some of this is joint work with Franziska Jahnke.

 

Montag, 10.12.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Salma Kuhlmann (Konstanz) 

The (valuative) difference rank of a difference field

(Work in progress with M. Matusinski and F. Point)

Abstract: There are several equivalent characterizations of the (valuative) rank of a valued field: via the chain of ideals of the valuation ring, the chain of convex subgroups of the value group, or the chain of final segments of the value set of the value group. The order type of any of the above described chains is the rank of the valued field.

This analysis can be extended to cases when the field admits extra structure. We considered for instance ordered exponential fields, introduced the (valuative) exponential rank, and gave a characterization completely analogous to the above, but taking into account the (extra) structure induced by the exponential map (on the ideals, convex subgroups and final segments).

In this talk, we push this analogy to the case of an (ordered) difference field and introduce the notion of "difference rank", naturally identifying it with the chain of fixed points of the automorphism induced on the (valuative) rank.

Freitag, 14.12.2012 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Sabine Burgdorf (Université de Neuchâtel)

(Gast von Markus Schweighofer)

Schmüdgen's fibre theorem for tracial moments

Abstract: The d-tracial moment problem considers the characterization of positive tracial linear forms on polynomials in noncommuting variables which have a representation via tracial moments of matrices of size d. For this dimension dependent tracial moment problem I present an analog of Schmüdgen's fibre theorem, where fibres will be generated by central polynomials on dxd matrices.

Montag, 17.12.2012 um 15.15 Uhr, Oberseminar Modelltheorie

Maurice Chiodo (Universita degli Studi di Milano)

(Gast von Salma Kuhlmann)

The computational complexity of recognising embeddings, and a universal finitely presented torsion-free group

Abstract: We extend a result by Lempp that recognising torsion-freeness for finitely presented groups is Π⁰₂-complete in the arithmetic hierarchy; we show that the problem of recognising embeddings of finitely presented groups is at least Π⁰₂-hard, Σ⁰₂-hard, and lies in Σ⁰₃. We give a uniform construction that, on input of a recursive presentation of a group P , outputs a recursive presentation P^tf of a torsion-free group which is isomorphic to P whenever P is itself torsion-free. We apply our constructions to form a ’universal’ finitely presented torsion-free group; one in which every finitely presented torsion-free group embeds.

Freitag, 21.12.2012 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

 

Montag, 07.01.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

Freitag, 11.01.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

 

Montag, 14.01.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Paola d'Aquino (Seconda Università degli Studi di Napoli)

(Gast von Salma Kuhlmann)

Saturation properties of o-minimal structures

Abstract: k-saturated divisible ordered abelian groups and real closed fields have been characterized in terms of valuation theory by S. Kuhlmann. I will present extensions of these results to o-minimal expansions of a real closed field and an analogous characterization of recursive saturation.

Freitag, 18.01.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Claus Scheiderer (Konstanz)

Sums of squares of polynomials with rational coefficients

http://cms.uni-konstanz.de/math/schwerpunkt-reelle-geometrie-und-algebra/vortraege-im-ws-201213-abstracts/#1712Abstract: We construct an explict polynomial with rational coefficients which is a sum of squares of polynomials with real coefficients, but not of polynomials with rational coefficients. Whether or not such examples exist was an open question originally raised by Sturmfels.

 

Montag, 21.01.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Lorna Gregory (Konstanz)

Integer parts and henselian ordered fields

Abstract: Shepherdson showed that models of open induction, that is discrete ordered rings whose positive parts satisfy induction of quantifier-free formulae, are exactly the integer parts of real closed fields. Mourgues and Ressayre showed that every real closed field has an integer part. Fornasiero extended this result to show that every ordered field which is henselian with respect to its smallest convex valuation has an integer part. We call such an ordered field a henselian ordered field. We show that the class of henselian ordered fields is not axiomatisable and that the class of iinteger parts of these fields is not axiomatisable either. We will then show that there is a largest axiomatisable class of henselian ordered fields and show that the class of integer parts of these ordered fields is axiomatisable. This largest axiomatisable class of henselian ordered fields turns out to be exactly the henselian ordered fields such that each of their elementary extensions is also a henselian ordered field and that as pure fields these are the almost real closed fields of Delon and Farré.

Freitag, 25.01.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

Daniel Plaumann (Konstanz)

Hyperbolic Polynomials and Interlacers

http://cms.uni-konstanz.de/math/schwerpunkt-reelle-geometrie-und-algebra/vortraege-im-ws-201213-abstracts/#1712Abstract: Hyperbolic polynomials are real polynomials with a simple reality condition on the zeros, reminiscent of determinants of symmetric matrices. Real polynomials whose zeros interlace a given hyperbolic polynomial come up in various places in the study of the hyperbolicity cone and determinantal representations. We will broadly discuss the role of these interlacers, as well as some recent results from joint work with Mario Kummer and Cynthia Vinzant.

 

Montag, 28.01.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Margaret Thomas (Konstanz)

Growth dichotomies for integer-valued definable functions

Abstract: We consider analytic functions which take integer values at natural number arguments, and address the question of their growth behaviour at infinity. We shall show that for functions definable in certain o-minimal expansions of the real field there are dichotomies for this behaviour in the direction of a classical theorem of P\'{o}lya (regarding complex entire functions). Our main result follows from work in a related area on a conjecture of Wilkie, which concerns the bounding of the density of rational and algebraic points on sets definable in the real exponential field (this conjecture puts forward an improvement to the influential counting theorem of Pila and Wilkie, a result which has already had far-reaching consequences for diophantine geometry). We will discuss the proofs of these growth dichotomies, explaining the necessary background along the way. This is joint work with Gareth O. Jones and Alex J. Wilkie.

Freitag, 01.02.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

Montag, 04.02.2013 um 15.15 Uhr, Oberseminar Modelltheorie

kein Vortrag

 

Freitag, 08.02.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

 

Montag, 11.02.2013 um 15.15 Uhr, Oberseminar Modelltheorie

Katharina Dupont (Konstanz)

The Cardinality of basis of V-topologies

Abstract: In my talk I will present some results on the cardinality of V-topologies partly based on my diploma thesis. I will examine (filter) basis of neighbourhoods of zero as well as basis in the topological sense and the connection between the two. For the special case of t-henselian fields I will sketch a proof why not every such field admits a countable basis of zero environments and give further conditions under which there exists one.
This work is motivated by a question which came up during a talk of Laurent Moret- Bailly in Paris in November 2012.

Freitag, 15.02.2013 um 13.30 Uhr, Oberseminar Reelle Geometrie und Algebra

kein Vortrag

 

Mittwoch, 13.03.2013 um 11.00 Uhr, Gastvortrag

Greg Blekherman (Georgia Tech)

(Gast von Daniel Plaumann)

A Tale of Two Theorems

Abstract: I will explain and draw connections between the following two theorems: (1) Hilbert's theorem on nonnegative polynomials and sums of squares, and (2) Classification of varieties of minimal degree by Del Pezzo and Bertini. This will result in the classification of all varieties on which nonnegative polynomials are equal to sums of squares. (Joint work with Greg Smith and Mauricio Velasco)